1) The document discusses several philosophical issues that arise at the foundations of quantum thermodynamics, including what constitutes measurement, the nature of probability and entropy, and how the classical world emerges from the quantum world. 2) It examines different conceptions of entropy, such as Boltzmann entropy based on macrostates and Gibbs/Shannon entropy based on probability distributions. A key question is whether entropy is objective or depends on what is known. 3) The arrow of time and why entropy increases over time is explored, as well as challenges like the reversibility objection and the role of initial conditions like the past hypothesis. The relationship between thermodynamics and cosmology is also discussed.

1. Philosophical Issues
in Quantum
Thermodynamics
Sean M. Carroll
Homewood Professor of Natural Philosophy,
Johns Hopkins
& Fractal Faculty, Santa Fe Institute
Dall-E

2. Philosophy, Physics, & Foundations
Philosophy Physics
Philosophy
of Physics
Foundations
of Physics
Physics Q: What is the probability
that A → B?
Philosophy Q: What is probability?
Foundations Q: Is the universe
fundamentally probabilistic?
Physicists w/
physics PhD
Philosophers w/
physics PhD
Philosophers w/
philosophy PhD

3. Pressing Foundational Questions
Quantum Mechanics:
• What constitutes “measurement”?
• Does the wave function represent reality,
or is it just a tool for making predictions?
• Are there hidden variables?
• What is the nature of non-locality?
• How does the classical world emerge?
Thermodynamics/Stat Mech:
• Is entropy objective or subjective?
• Why does entropy increase?
• How/when do systems equilibrate?
• Where do probabilities come from?
• What is the role of information?
Plus:
Foundations of
classical mechanics,
quantum field theory,
spacetime,
cosmology,
mathematics,
biology, economics...

4. Versions of Entropy
“Boltzmann”: Coarse-grain into macrostates.
Entropy = log of macrostate volume, .
space of
states
“Gibbs/Shannon”: Probability distribution
over microstates.
<latexit sha1_base64="7s8Q65UOtPMz81pIvF2oQuAZFuA=">AAACAXicbZDLSsNAFIYnXmu9Rd0IbgaL4MaSiJduhIIblxXtBZoQJtNJO3RmEmYmQgl146u4caGIW9/CnW/jpM1CW3848PGfc5g5f5gwqrTjfFsLi0vLK6ultfL6xubWtr2z21JxKjFp4pjFshMiRRgVpKmpZqSTSIJ4yEg7HF7n/fYDkYrG4l6PEuJz1Bc0ohhpYwX2/h28gieeSnlAYWLKY3E/h8CuOFVnIjgPbgEVUKgR2F9eL8YpJ0JjhpTquk6i/QxJTTEj47KXKpIgPER90jUoECfKzyYXjOGRcXowiqUpoeHE/b2RIa7UiIdmkiM9ULO93Pyv1011VPMzKpJUE4GnD0UpgzqGeRywRyXBmo0MICyp+SvEAyQR1ia0sgnBnT15HlqnVfeien57VqnXijhK4AAcgmPggktQBzegAZoAg0fwDF7Bm/VkvVjv1sd0dMEqdvbAH1mfP0bklX8=</latexit>
S =
X
i
pi log pi
Is entropy an objective feature of a system, or
a characteristic of what we know?
Myrvold’s question.
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k log W

5. Statistics, Ergodicity & Equilibration
• Systems approach equilibrium. Is that a law?
• Ergodicity: time-averages equal state-averages.
Systems spend most of their time in equilibrium.
But: many systems not ergodic.
And timescales aren’t really infinite.
• We assume systems are in a “typical” microstate with respect to
macroscopic constraints. But why?
(We know they aren’t, really: just evolve backwards.)
• Bottom line: we know assumptions from which thermodynamics
can be derived, but are they the right ones?

6. The Arrow of Time
<latexit sha1_base64="eP+yK7ZpNMR/XQcb7aWNGazZsS8=">AAAB/XicdVDLSsNAFJ34rPUVHzs3g0VwFSatqXVXcOOyon1AE8pkMmmHTh7OTIQair/ixoUibv0Pd/6N04egogcuHM65l3vv8VPOpELow1hYXFpeWS2sFdc3Nre2zZ3dlkwyQWiTJDwRHR9LyllMm4opTjupoDjyOW37w/OJ376lQrIkvlajlHoR7scsZAQrLfXMfTcUmOTB1TgP1Bi6nN5A1DNLyEJVp3LmQGQ5yK6VJ6Ts2AhVoG2hKUpgjkbPfHeDhGQRjRXhWMqujVLl5VgoRjgdF91M0hSTIe7TrqYxjqj08un1Y3iklQCGidAVKzhVv0/kOJJyFPm6M8JqIH97E/Evr5upsOblLE4zRWMyWxRmHKoETqKAAROUKD7SBBPB9K2QDLCOQ+nAijqEr0/h/6RVtuyq5VyelOq1eRwFcAAOwTGwwSmogwvQAE1AwB14AE/g2bg3Ho0X43XWumDMZ/bADxhvn3FGlTQ=</latexit>
dS
dt
 0
Boltzmann: Entropy tends to increase
because there are more ways to be
high-entropy than low-entropy
Lohschmidt (“reversibility objection”):
But there are equal number of
trajectories that go high → low as there
are trajectories that go low → high.
Resolution: break time-symmetry by
asserting a Past Hypothesis – the
universe “started” in a low entropy
state some finite time ago.
Big
Bang
[Albert 2000]

7. 1 sec 105 yr 1010 yr 1015 yr 10100 yr
Why did the early universe have low entropy?
Thermodynamics relies on cosmology

8. Measurements & the Second Law
Gibbs/Shannon entropy:
<latexit sha1_base64="7s8Q65UOtPMz81pIvF2oQuAZFuA=">AAACAXicbZDLSsNAFIYnXmu9Rd0IbgaL4MaSiJduhIIblxXtBZoQJtNJO3RmEmYmQgl146u4caGIW9/CnW/jpM1CW3848PGfc5g5f5gwqrTjfFsLi0vLK6ultfL6xubWtr2z21JxKjFp4pjFshMiRRgVpKmpZqSTSIJ4yEg7HF7n/fYDkYrG4l6PEuJz1Bc0ohhpYwX2/h28gieeSnlAYWLKY3E/h8CuOFVnIjgPbgEVUKgR2F9eL8YpJ0JjhpTquk6i/QxJTTEj47KXKpIgPER90jUoECfKzyYXjOGRcXowiqUpoeHE/b2RIa7UiIdmkiM9ULO93Pyv1011VPMzKpJUE4GnD0UpgzqGeRywRyXBmo0MICyp+SvEAyQR1ia0sgnBnT15HlqnVfeien57VqnXijhK4AAcgmPggktQBzegAZoAg0fwDF7Bm/VkvVjv1sd0dMEqdvbAH1mfP0bklX8=</latexit>
S =
X
i
pi log pi
In a closed system, strictly constant. Upon measurement, decreases!
What happened to the Second Law?
HC[ , ] = (x) log (x) dx = I
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Cross-entropy: likelihood of surprise if you’re using “wrong” distribution.
Bayesian Second law: after a measurement outcome m, the cross-entropy
between original distribution and measurement-updated distribution is
higher than it would have been initially.
HC[ m, ] 0
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[Bartolotta, Carroll, Leichenauer & Pollack]

9. Maxwell’s Demon
Note: reliable measurement
requires an arrow of time.
Would be nice to
understand better. A A
A
A A
B
Landauer/Bennett: it creates k log(2) to
erase a bit of the Demon’s memory.
Earman & Norton: that’s only if you
assume the 2nd law from the start.
Still a controversy?
?

10. Quantum Mechanics: Foundational Theories
Does QM describe
reality directly, or
simply predict
observational outcomes?
Is the wave function
the whole story, or
are there other
physical quantities?
Does the wave function
always obey the
Schrödinger equation,
or does it truly collapse?
Everett/
Many-Worlds
Objective
Collapse
Models
Hidden
Variables
(de Broglie/
Bohm)
Epistemic
Models
(Copenhagen,
QBism)
predict
reality
others
only
𝚿
Schrödinger
collapse

11. Measurement & Energy Conservation
This is manifestly not conserved during quantum measurement
in the conventional picture.
Define energy in a state as
| i E = h |Ĥ| i.
| i
| i
| i
+
| i
[Pearle; Aharonov et al; Maudlin, Okon, & Sudarsky; Sołtan et al.; Carroll & Lodman]
You might suspect that energy is transferred to the apparatus or
environment, but it’s easy to show that can’t restore conservation
in general. Potentially experimentally observable.
Does that bother us? Should it?

12. Decoherence & Pointer States
photon
[Zurek et al.]
Quantum states of macroscopic
systems collapse (objectively
or seemingly) onto some states and
not others: classical-looking
pointer states.
Why those?
Key role is played by locality. Interactions are local in space.
Hence, spatially-incoherent quantum states quickly decohere.
Why are the laws of physics local? Could it have been different?

13. How to Partition Hilbert Space?
(Quantum Mereology)
[Carroll & Singh 2021]
Ĥ = ĤS ⌦ IE + IS ⌦ ĤE + ĤI.
The algorithm is: given , sift through all possible
factorizations that simultaneously minimize
spread of the system wave function and
entanglement with the environment.
Ĥ
HS ⌦ HE
HE
HS
System Hamiltonian
governs
“remains localized”
Interaction Hamiltonian
governs
“remains unentangled”
What physical principle tells us the “best” way to
decompose Hilbert space, e.g. into system/environment?

14. Cosmology Revisited
1 sec 105 yr 1010 yr 1015 yr 10100 yr
The quantum arrow of time
(decoherence/branching/collapse)
has the same origin as the ordinary
thermodynamic arrow: coarse-graining
+ low-entropy past hypothesis.
The Big Bang is the ultimate quantum thermodynamic resource.
Without it, we’d live in thermodynamic equilibrium.
(Except there would be no life.)
Probably need quantum cosmology to understand
why our universe is like that. Is it just a fluctuation?
[Carroll & Chen 2004]

15. Boltzmann Brains
If the universe is eternal, and undergoes thermal
fluctuations, it will produce an infinite number of
minimal random observers – “Boltzmann Brains.”
Standard cosmology (LCDM): our
universe has a horizon, and the
quantum state inside settles into a
thermal density matrix with T = 10-37 K.
horizon
[Boddy, Carroll & Pollack 2014 vs. Lloyd 2016]
Lingering question: does a (static!) thermal density matrix
exhibit real, dynamical fluctuations? Real enough to be ”alive”?
<latexit sha1_base64="LuJngoGXhDY1S+vWsuYZ4WRl3M8=">AAACIXicbVBNSyNBEO3xY9Wou1k9itBsELwYZoTVXBYCsugxwiYKmTjUdCpJY0/P0F0jhDF/xYtXf8Z68OCy5Cb+me18HHZ1H3TX470quuvFmZKWfP/FW1hcWv6wsrpWWt/Y/Pip/HmrZdPcCGyKVKXmMgaLSmpskiSFl5lBSGKFF/H1ycS/uEFjZap/0DDDTgJ9LXtSADkpKtfCAVBoBin/xkObJ5HmeFUchDES8O+RHt26KzSg+wp5qGbVSbdRueJX/Sn4exLMSaW+e/r4cLpz1ojK47CbijxBTUKBte3Az6hTgCEpFI5KYW4xA3ENfWw7qiFB2ymmG474nlO6vJcadzTxqfr3RAGJtcMkdp0J0MC+9Sbi/7x2Tr1ap5A6ywm1mD3UyxWnlE/i4l1pUJAaOgLCSPdXLgZgQJALteRCCN6u/J60DqvBUfXreVCp19gMq2yHfWH7LGDHrM7OWIM1mWB37Cd7Zr+8e+/J++2NZ60L3nxmm/0D7/UPeGOmAg==</latexit>
ˆ
⇢ =
P
n e En
|EnihEn|

16. Foundational Questions for Quantum Thermodynamics
• What are the best definitions of work/energy/entropy/information?
• What are measurements (including ”weak”)?
• What is the role of an environment/heat bath?
• How does entanglement contribute to equilibration?
• What does it mean to turn information into work?
• What is the role of probability & quasi-probability?
• In what senses do quantum systems “fluctuate”?
• Why and how do systems become classical?
• Are there fundamental limits on knowledge/information processing?
• What is the role of measurements/interventions?
• Is quantum thermodynamics important for complexity?
• Can QM help us understand the foundations of thermodynamics?